Some exact solutions to the nonlinear shallow-water wave equations

Abstract

These exact solutions correspond to time-dependent motions in parabolic basins. A characteristic feature is that the shoreline is not fixed. It is free to move and must be determined as part of the solution. In general, the motion is oscillatory and has the appropriate small-amplitude limit. For the case in which the parabolic basin reduces to a flat plane, there is a solution for a flood wave. These solutions provide a valuable test for numerical models of inundating storm tides.

SWASHES: a compilation of shallow water analytic solutions for hydraulic and environmental studies

“…Most of them include the Coriolis force that we do not consider here (for further information, see [48]). These solutions are periodic in time with moving wet/dry transitions.…”

“…The free surface has a periodic motion and remains planar in time [48]. To visualize this case, one can think of a glass with some liquid in rotation inside.…”

A well‐balanced Runge–Kutta discontinuous Galerkin method for the shallow‐water equations with flooding and drying

“…We consider a parabolic bowl (the bottom corresponds to a paraboloid of revolution, i.e., b(x, y) = αr 2 with r 2 = x 2 + y 2 and α is a positive constant) for which the exact solution has a periodic behavior and the free surface is an oscillating paraboloid of revolution. The analytical solution (see [35] for more details) is such that ζ(r, t) is non-zero for r < X+Y cos ωt (with ω 2 = 8gα, X and Y are constants such that X > 0 and |Y | < X), and…”

Modeling wave runup with depth-integrated equations

“…One such solution is that for a long wave resonating in a circular parabolic basin. Thacker (1981) presented a solution to the NLSW equations, where the initial free surface displacement is given as:…”